3.22 \(\int \frac{\sinh ^3(a+b x)}{(c+d x)^3} \, dx\)

Optimal. Leaf size=184 \[ \frac{9 b^2 \sinh \left (3 a-\frac{3 b c}{d}\right ) \text{Chi}\left (\frac{3 b c}{d}+3 b x\right )}{8 d^3}-\frac{3 b^2 \sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{8 d^3}-\frac{3 b^2 \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{8 d^3}+\frac{9 b^2 \cosh \left (3 a-\frac{3 b c}{d}\right ) \text{Shi}\left (\frac{3 b c}{d}+3 b x\right )}{8 d^3}-\frac{3 b \sinh ^2(a+b x) \cosh (a+b x)}{2 d^2 (c+d x)}-\frac{\sinh ^3(a+b x)}{2 d (c+d x)^2} \]

[Out]

(9*b^2*CoshIntegral[(3*b*c)/d + 3*b*x]*Sinh[3*a - (3*b*c)/d])/(8*d^3) - (3*b^2*CoshIntegral[(b*c)/d + b*x]*Sin
h[a - (b*c)/d])/(8*d^3) - (3*b*Cosh[a + b*x]*Sinh[a + b*x]^2)/(2*d^2*(c + d*x)) - Sinh[a + b*x]^3/(2*d*(c + d*
x)^2) - (3*b^2*Cosh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/(8*d^3) + (9*b^2*Cosh[3*a - (3*b*c)/d]*SinhInteg
ral[(3*b*c)/d + 3*b*x])/(8*d^3)

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Rubi [A]  time = 0.424843, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {3314, 3303, 3298, 3301, 3312} \[ \frac{9 b^2 \sinh \left (3 a-\frac{3 b c}{d}\right ) \text{Chi}\left (\frac{3 b c}{d}+3 b x\right )}{8 d^3}-\frac{3 b^2 \sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{8 d^3}-\frac{3 b^2 \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{8 d^3}+\frac{9 b^2 \cosh \left (3 a-\frac{3 b c}{d}\right ) \text{Shi}\left (\frac{3 b c}{d}+3 b x\right )}{8 d^3}-\frac{3 b \sinh ^2(a+b x) \cosh (a+b x)}{2 d^2 (c+d x)}-\frac{\sinh ^3(a+b x)}{2 d (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x]^3/(c + d*x)^3,x]

[Out]

(9*b^2*CoshIntegral[(3*b*c)/d + 3*b*x]*Sinh[3*a - (3*b*c)/d])/(8*d^3) - (3*b^2*CoshIntegral[(b*c)/d + b*x]*Sin
h[a - (b*c)/d])/(8*d^3) - (3*b*Cosh[a + b*x]*Sinh[a + b*x]^2)/(2*d^2*(c + d*x)) - Sinh[a + b*x]^3/(2*d*(c + d*
x)^2) - (3*b^2*Cosh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/(8*d^3) + (9*b^2*Cosh[3*a - (3*b*c)/d]*SinhInteg
ral[(3*b*c)/d + 3*b*x])/(8*d^3)

Rule 3314

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(b*Si
n[e + f*x])^n)/(d*(m + 1)), x] + (Dist[(b^2*f^2*n*(n - 1))/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin
[e + f*x])^(n - 2), x], x] - Dist[(f^2*n^2)/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x
], x] - Simp[(b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1))/(d^2*(m + 1)*(m + 2)), x]) /; Fre
eQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rubi steps

\begin{align*} \int \frac{\sinh ^3(a+b x)}{(c+d x)^3} \, dx &=-\frac{3 b \cosh (a+b x) \sinh ^2(a+b x)}{2 d^2 (c+d x)}-\frac{\sinh ^3(a+b x)}{2 d (c+d x)^2}+\frac{\left (3 b^2\right ) \int \frac{\sinh (a+b x)}{c+d x} \, dx}{d^2}+\frac{\left (9 b^2\right ) \int \frac{\sinh ^3(a+b x)}{c+d x} \, dx}{2 d^2}\\ &=-\frac{3 b \cosh (a+b x) \sinh ^2(a+b x)}{2 d^2 (c+d x)}-\frac{\sinh ^3(a+b x)}{2 d (c+d x)^2}+\frac{\left (9 i b^2\right ) \int \left (\frac{3 i \sinh (a+b x)}{4 (c+d x)}-\frac{i \sinh (3 a+3 b x)}{4 (c+d x)}\right ) \, dx}{2 d^2}+\frac{\left (3 b^2 \cosh \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sinh \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx}{d^2}+\frac{\left (3 b^2 \sinh \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cosh \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx}{d^2}\\ &=\frac{3 b^2 \text{Chi}\left (\frac{b c}{d}+b x\right ) \sinh \left (a-\frac{b c}{d}\right )}{d^3}-\frac{3 b \cosh (a+b x) \sinh ^2(a+b x)}{2 d^2 (c+d x)}-\frac{\sinh ^3(a+b x)}{2 d (c+d x)^2}+\frac{3 b^2 \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{d^3}+\frac{\left (9 b^2\right ) \int \frac{\sinh (3 a+3 b x)}{c+d x} \, dx}{8 d^2}-\frac{\left (27 b^2\right ) \int \frac{\sinh (a+b x)}{c+d x} \, dx}{8 d^2}\\ &=\frac{3 b^2 \text{Chi}\left (\frac{b c}{d}+b x\right ) \sinh \left (a-\frac{b c}{d}\right )}{d^3}-\frac{3 b \cosh (a+b x) \sinh ^2(a+b x)}{2 d^2 (c+d x)}-\frac{\sinh ^3(a+b x)}{2 d (c+d x)^2}+\frac{3 b^2 \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{d^3}+\frac{\left (9 b^2 \cosh \left (3 a-\frac{3 b c}{d}\right )\right ) \int \frac{\sinh \left (\frac{3 b c}{d}+3 b x\right )}{c+d x} \, dx}{8 d^2}-\frac{\left (27 b^2 \cosh \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sinh \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx}{8 d^2}+\frac{\left (9 b^2 \sinh \left (3 a-\frac{3 b c}{d}\right )\right ) \int \frac{\cosh \left (\frac{3 b c}{d}+3 b x\right )}{c+d x} \, dx}{8 d^2}-\frac{\left (27 b^2 \sinh \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cosh \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx}{8 d^2}\\ &=\frac{9 b^2 \text{Chi}\left (\frac{3 b c}{d}+3 b x\right ) \sinh \left (3 a-\frac{3 b c}{d}\right )}{8 d^3}-\frac{3 b^2 \text{Chi}\left (\frac{b c}{d}+b x\right ) \sinh \left (a-\frac{b c}{d}\right )}{8 d^3}-\frac{3 b \cosh (a+b x) \sinh ^2(a+b x)}{2 d^2 (c+d x)}-\frac{\sinh ^3(a+b x)}{2 d (c+d x)^2}-\frac{3 b^2 \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{8 d^3}+\frac{9 b^2 \cosh \left (3 a-\frac{3 b c}{d}\right ) \text{Shi}\left (\frac{3 b c}{d}+3 b x\right )}{8 d^3}\\ \end{align*}

Mathematica [A]  time = 0.842618, size = 220, normalized size = 1.2 \[ \frac{6 b^2 (c+d x)^2 \left (3 \sinh \left (3 a-\frac{3 b c}{d}\right ) \text{Chi}\left (\frac{3 b (c+d x)}{d}\right )-\sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (b \left (\frac{c}{d}+x\right )\right )-\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (b \left (\frac{c}{d}+x\right )\right )+3 \cosh \left (3 a-\frac{3 b c}{d}\right ) \text{Shi}\left (\frac{3 b (c+d x)}{d}\right )\right )+6 d \cosh (b x) (b \cosh (a) (c+d x)+d \sinh (a))-2 d \cosh (3 b x) (3 b \cosh (3 a) (c+d x)+d \sinh (3 a))+6 d \sinh (b x) (b \sinh (a) (c+d x)+d \cosh (a))-2 d \sinh (3 b x) (3 b \sinh (3 a) (c+d x)+d \cosh (3 a))}{16 d^3 (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*x]^3/(c + d*x)^3,x]

[Out]

(6*d*Cosh[b*x]*(b*(c + d*x)*Cosh[a] + d*Sinh[a]) - 2*d*Cosh[3*b*x]*(3*b*(c + d*x)*Cosh[3*a] + d*Sinh[3*a]) + 6
*d*(d*Cosh[a] + b*(c + d*x)*Sinh[a])*Sinh[b*x] - 2*d*(d*Cosh[3*a] + 3*b*(c + d*x)*Sinh[3*a])*Sinh[3*b*x] + 6*b
^2*(c + d*x)^2*(3*CoshIntegral[(3*b*(c + d*x))/d]*Sinh[3*a - (3*b*c)/d] - CoshIntegral[b*(c/d + x)]*Sinh[a - (
b*c)/d] - Cosh[a - (b*c)/d]*SinhIntegral[b*(c/d + x)] + 3*Cosh[3*a - (3*b*c)/d]*SinhIntegral[(3*b*(c + d*x))/d
]))/(16*d^3*(c + d*x)^2)

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Maple [B]  time = 0.089, size = 562, normalized size = 3.1 \begin{align*} -{\frac{3\,{b}^{3}{{\rm e}^{-3\,bx-3\,a}}x}{16\,d \left ({b}^{2}{d}^{2}{x}^{2}+2\,{b}^{2}cdx+{c}^{2}{b}^{2} \right ) }}-{\frac{3\,{b}^{3}{{\rm e}^{-3\,bx-3\,a}}c}{16\,{d}^{2} \left ({b}^{2}{d}^{2}{x}^{2}+2\,{b}^{2}cdx+{c}^{2}{b}^{2} \right ) }}+{\frac{{b}^{2}{{\rm e}^{-3\,bx-3\,a}}}{16\,d \left ({b}^{2}{d}^{2}{x}^{2}+2\,{b}^{2}cdx+{c}^{2}{b}^{2} \right ) }}+{\frac{9\,{b}^{2}}{16\,{d}^{3}}{{\rm e}^{-3\,{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,3\,bx+3\,a-3\,{\frac{da-cb}{d}} \right ) }+{\frac{3\,{b}^{3}{{\rm e}^{-bx-a}}x}{16\,d \left ({b}^{2}{d}^{2}{x}^{2}+2\,{b}^{2}cdx+{c}^{2}{b}^{2} \right ) }}+{\frac{3\,{b}^{3}{{\rm e}^{-bx-a}}c}{16\,{d}^{2} \left ({b}^{2}{d}^{2}{x}^{2}+2\,{b}^{2}cdx+{c}^{2}{b}^{2} \right ) }}-{\frac{3\,{b}^{2}{{\rm e}^{-bx-a}}}{16\,d \left ({b}^{2}{d}^{2}{x}^{2}+2\,{b}^{2}cdx+{c}^{2}{b}^{2} \right ) }}-{\frac{3\,{b}^{2}}{16\,{d}^{3}}{{\rm e}^{-{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{da-cb}{d}} \right ) }+{\frac{3\,{b}^{2}{{\rm e}^{bx+a}}}{16\,{d}^{3}} \left ({\frac{cb}{d}}+bx \right ) ^{-2}}+{\frac{3\,{b}^{2}{{\rm e}^{bx+a}}}{16\,{d}^{3}} \left ({\frac{cb}{d}}+bx \right ) ^{-1}}+{\frac{3\,{b}^{2}}{16\,{d}^{3}}{{\rm e}^{{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,-bx-a-{\frac{-da+cb}{d}} \right ) }-{\frac{{b}^{2}{{\rm e}^{3\,bx+3\,a}}}{16\,{d}^{3}} \left ({\frac{cb}{d}}+bx \right ) ^{-2}}-{\frac{3\,{b}^{2}{{\rm e}^{3\,bx+3\,a}}}{16\,{d}^{3}} \left ({\frac{cb}{d}}+bx \right ) ^{-1}}-{\frac{9\,{b}^{2}}{16\,{d}^{3}}{{\rm e}^{3\,{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,-3\,bx-3\,a-3\,{\frac{-da+cb}{d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(b*x+a)^3/(d*x+c)^3,x)

[Out]

-3/16*b^3*exp(-3*b*x-3*a)/d/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)*x-3/16*b^3*exp(-3*b*x-3*a)/d^2/(b^2*d^2*x^2+2*b^
2*c*d*x+b^2*c^2)*c+1/16*b^2*exp(-3*b*x-3*a)/d/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)+9/16*b^2/d^3*exp(-3*(a*d-b*c)/
d)*Ei(1,3*b*x+3*a-3*(a*d-b*c)/d)+3/16*b^3*exp(-b*x-a)/d/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)*x+3/16*b^3*exp(-b*x-
a)/d^2/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)*c-3/16*b^2*exp(-b*x-a)/d/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)-3/16*b^2/d
^3*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)+3/16*b^2/d^3*exp(b*x+a)/(b*c/d+b*x)^2+3/16*b^2/d^3*exp(b*x+a)/(b*
c/d+b*x)+3/16*b^2/d^3*exp((a*d-b*c)/d)*Ei(1,-b*x-a-(-a*d+b*c)/d)-1/16*b^2/d^3*exp(3*b*x+3*a)/(b*c/d+b*x)^2-3/1
6*b^2/d^3*exp(3*b*x+3*a)/(b*c/d+b*x)-9/16*b^2/d^3*exp(3*(a*d-b*c)/d)*Ei(1,-3*b*x-3*a-3*(-a*d+b*c)/d)

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Maxima [A]  time = 1.60042, size = 196, normalized size = 1.07 \begin{align*} \frac{e^{\left (-3 \, a + \frac{3 \, b c}{d}\right )} E_{3}\left (\frac{3 \,{\left (d x + c\right )} b}{d}\right )}{8 \,{\left (d x + c\right )}^{2} d} - \frac{3 \, e^{\left (-a + \frac{b c}{d}\right )} E_{3}\left (\frac{{\left (d x + c\right )} b}{d}\right )}{8 \,{\left (d x + c\right )}^{2} d} + \frac{3 \, e^{\left (a - \frac{b c}{d}\right )} E_{3}\left (-\frac{{\left (d x + c\right )} b}{d}\right )}{8 \,{\left (d x + c\right )}^{2} d} - \frac{e^{\left (3 \, a - \frac{3 \, b c}{d}\right )} E_{3}\left (-\frac{3 \,{\left (d x + c\right )} b}{d}\right )}{8 \,{\left (d x + c\right )}^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)^3/(d*x+c)^3,x, algorithm="maxima")

[Out]

1/8*e^(-3*a + 3*b*c/d)*exp_integral_e(3, 3*(d*x + c)*b/d)/((d*x + c)^2*d) - 3/8*e^(-a + b*c/d)*exp_integral_e(
3, (d*x + c)*b/d)/((d*x + c)^2*d) + 3/8*e^(a - b*c/d)*exp_integral_e(3, -(d*x + c)*b/d)/((d*x + c)^2*d) - 1/8*
e^(3*a - 3*b*c/d)*exp_integral_e(3, -3*(d*x + c)*b/d)/((d*x + c)^2*d)

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Fricas [B]  time = 2.66529, size = 1123, normalized size = 6.1 \begin{align*} -\frac{2 \, d^{2} \sinh \left (b x + a\right )^{3} + 6 \,{\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right )^{3} + 18 \,{\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 6 \,{\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right ) + 3 \,{\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )}{\rm Ei}\left (\frac{b d x + b c}{d}\right ) -{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )}{\rm Ei}\left (-\frac{b d x + b c}{d}\right )\right )} \cosh \left (-\frac{b c - a d}{d}\right ) - 9 \,{\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )}{\rm Ei}\left (\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) -{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )}{\rm Ei}\left (-\frac{3 \,{\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right ) + 6 \,{\left (d^{2} \cosh \left (b x + a\right )^{2} - d^{2}\right )} \sinh \left (b x + a\right ) + 3 \,{\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )}{\rm Ei}\left (\frac{b d x + b c}{d}\right ) +{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )}{\rm Ei}\left (-\frac{b d x + b c}{d}\right )\right )} \sinh \left (-\frac{b c - a d}{d}\right ) - 9 \,{\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )}{\rm Ei}\left (\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) +{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )}{\rm Ei}\left (-\frac{3 \,{\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right )}{16 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)^3/(d*x+c)^3,x, algorithm="fricas")

[Out]

-1/16*(2*d^2*sinh(b*x + a)^3 + 6*(b*d^2*x + b*c*d)*cosh(b*x + a)^3 + 18*(b*d^2*x + b*c*d)*cosh(b*x + a)*sinh(b
*x + a)^2 - 6*(b*d^2*x + b*c*d)*cosh(b*x + a) + 3*((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*Ei((b*d*x + b*c)/d) -
(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*Ei(-(b*d*x + b*c)/d))*cosh(-(b*c - a*d)/d) - 9*((b^2*d^2*x^2 + 2*b^2*c*
d*x + b^2*c^2)*Ei(3*(b*d*x + b*c)/d) - (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*Ei(-3*(b*d*x + b*c)/d))*cosh(-3*(
b*c - a*d)/d) + 6*(d^2*cosh(b*x + a)^2 - d^2)*sinh(b*x + a) + 3*((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*Ei((b*d
*x + b*c)/d) + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*Ei(-(b*d*x + b*c)/d))*sinh(-(b*c - a*d)/d) - 9*((b^2*d^2*
x^2 + 2*b^2*c*d*x + b^2*c^2)*Ei(3*(b*d*x + b*c)/d) + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*Ei(-3*(b*d*x + b*c)
/d))*sinh(-3*(b*c - a*d)/d))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)**3/(d*x+c)**3,x)

[Out]

Integral(sinh(a + b*x)**3/(c + d*x)**3, x)

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Giac [B]  time = 1.18637, size = 811, normalized size = 4.41 \begin{align*} \frac{9 \, b^{2} d^{2} x^{2}{\rm Ei}\left (\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (3 \, a - \frac{3 \, b c}{d}\right )} - 3 \, b^{2} d^{2} x^{2}{\rm Ei}\left (\frac{b d x + b c}{d}\right ) e^{\left (a - \frac{b c}{d}\right )} + 3 \, b^{2} d^{2} x^{2}{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (-a + \frac{b c}{d}\right )} - 9 \, b^{2} d^{2} x^{2}{\rm Ei}\left (-\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac{3 \, b c}{d}\right )} + 18 \, b^{2} c d x{\rm Ei}\left (\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (3 \, a - \frac{3 \, b c}{d}\right )} - 6 \, b^{2} c d x{\rm Ei}\left (\frac{b d x + b c}{d}\right ) e^{\left (a - \frac{b c}{d}\right )} + 6 \, b^{2} c d x{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (-a + \frac{b c}{d}\right )} - 18 \, b^{2} c d x{\rm Ei}\left (-\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac{3 \, b c}{d}\right )} + 9 \, b^{2} c^{2}{\rm Ei}\left (\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (3 \, a - \frac{3 \, b c}{d}\right )} - 3 \, b^{2} c^{2}{\rm Ei}\left (\frac{b d x + b c}{d}\right ) e^{\left (a - \frac{b c}{d}\right )} + 3 \, b^{2} c^{2}{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (-a + \frac{b c}{d}\right )} - 9 \, b^{2} c^{2}{\rm Ei}\left (-\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac{3 \, b c}{d}\right )} - 3 \, b d^{2} x e^{\left (3 \, b x + 3 \, a\right )} + 3 \, b d^{2} x e^{\left (b x + a\right )} + 3 \, b d^{2} x e^{\left (-b x - a\right )} - 3 \, b d^{2} x e^{\left (-3 \, b x - 3 \, a\right )} - 3 \, b c d e^{\left (3 \, b x + 3 \, a\right )} + 3 \, b c d e^{\left (b x + a\right )} + 3 \, b c d e^{\left (-b x - a\right )} - 3 \, b c d e^{\left (-3 \, b x - 3 \, a\right )} - d^{2} e^{\left (3 \, b x + 3 \, a\right )} + 3 \, d^{2} e^{\left (b x + a\right )} - 3 \, d^{2} e^{\left (-b x - a\right )} + d^{2} e^{\left (-3 \, b x - 3 \, a\right )}}{16 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)^3/(d*x+c)^3,x, algorithm="giac")

[Out]

1/16*(9*b^2*d^2*x^2*Ei(3*(b*d*x + b*c)/d)*e^(3*a - 3*b*c/d) - 3*b^2*d^2*x^2*Ei((b*d*x + b*c)/d)*e^(a - b*c/d)
+ 3*b^2*d^2*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 9*b^2*d^2*x^2*Ei(-3*(b*d*x + b*c)/d)*e^(-3*a + 3*b*c/d)
+ 18*b^2*c*d*x*Ei(3*(b*d*x + b*c)/d)*e^(3*a - 3*b*c/d) - 6*b^2*c*d*x*Ei((b*d*x + b*c)/d)*e^(a - b*c/d) + 6*b^2
*c*d*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 18*b^2*c*d*x*Ei(-3*(b*d*x + b*c)/d)*e^(-3*a + 3*b*c/d) + 9*b^2*c^
2*Ei(3*(b*d*x + b*c)/d)*e^(3*a - 3*b*c/d) - 3*b^2*c^2*Ei((b*d*x + b*c)/d)*e^(a - b*c/d) + 3*b^2*c^2*Ei(-(b*d*x
+ b*c)/d)*e^(-a + b*c/d) - 9*b^2*c^2*Ei(-3*(b*d*x + b*c)/d)*e^(-3*a + 3*b*c/d) - 3*b*d^2*x*e^(3*b*x + 3*a) +
3*b*d^2*x*e^(b*x + a) + 3*b*d^2*x*e^(-b*x - a) - 3*b*d^2*x*e^(-3*b*x - 3*a) - 3*b*c*d*e^(3*b*x + 3*a) + 3*b*c*
d*e^(b*x + a) + 3*b*c*d*e^(-b*x - a) - 3*b*c*d*e^(-3*b*x - 3*a) - d^2*e^(3*b*x + 3*a) + 3*d^2*e^(b*x + a) - 3*
d^2*e^(-b*x - a) + d^2*e^(-3*b*x - 3*a))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)